Equivariant cohomology of cohomogeneity one actions: the topological case

We show that for any cohomogeneity one continuous action of a compact connected Lie group $G$ on a closed topological manifold the equivariant cohomology equipped with its canonical $H^*(BG)$-module structure is Cohen-Macaulay. The proof relies on the structure theorem for these actions recently obtained by Galaz-Garcia and Zarei. We generalize in this way our previous result concerning smooth actions.